All Kähler metrics on a complex manifold?

adding my comment as an answer

In general, Kahler metrics in $[ω_0]$ can also be parametrised as metrics of the same volume conformally equivalent to $ω_0$ by

$$\{\varphi\in C^\infty(X,\mathbb R)|\; \int_Xe^\varphi\omega_0^n=\int_X\omega_0^n=vol(X,[\omega_0]) \}$$

Moreover if two metric be comformally equivalent $\omega_\varphi^n=e^u\omega_0^n$ then conformal factor and Kahler potential are related by $(1+\Delta_{\omega_0}\varphi)=e^u$

In Mirror symmetric language

If $X$ and $\hat X$ be mirror to each other and be CY, then the Kahler moduli space of $\hat X$, denoted by $\mathcal M_{kah}(\hat X)$ can be identified with $K_\mathbb C(\hat X)/Aut(\hat X)$ where $$K_\mathbb C(\hat X)=\{\omega\in H^2(\hat X,\mathbb C)|Im(\omega)\in K(\hat X)\}/im H^2(\hat X,\mathbb Z)$$

We can identify the moduli space of Kahler spaces of $\hat X$ with moduli space of complex space of $X$ via Yukawa couplings

See Moduli Spaces of Hyperkahler Manifolds and Mirror Symmetry by Daniel Huybrechts

Let $M$ be a compact complex manifold. The set of Kahler metric on $M$ in a given Kahler class is parametrized by the set of positive volume forms with given integral, because (by Calabi-Yau theorem) any given positive volume form is a volume form of a Kahler metric in a given cohomology class, assuming their integrals agree. This is actually used when they put a structure of an infinite-dimensional symmetric space on the space of all Kahler metrics. See for example here: http://www.emis.de/journals/NYJM/JDG/p/2000/56-2-1.pdf (THE SPACE OF KAHLER METRICS, by XIUXIONG CHEN).

Compact case (since you mention $\mathbf C^2/\mathbf Z^4$):

• For $M$ to be Kähler its first Betti number must be even, and conversely every compact complex surface with even $b_1(M)$ is Kähler (Kodaira's conjecture, proved by Siu, Lamari, Buchdahl).

• Lamari and Buchdahl also describe "how many" Kähler metrics then exist, i.e. the so-called "Kähler cone" of classes in $H^{1,1}_{\mathbf R}(M)$ which can be represented by positive closed $(1,1)$-forms.